In Problems 59 - 62 , discuss the validity of each statement. If the r statement is true, explain why. If not, give a counterexample. For each positive integer n , the sum of the series 1 2 + 1 4 + 1 8 − ⋯ + − 1 n + 1 2 n is greater than or equal to 1 4 .
In Problems 59 - 62 , discuss the validity of each statement. If the r statement is true, explain why. If not, give a counterexample. For each positive integer n , the sum of the series 1 2 + 1 4 + 1 8 − ⋯ + − 1 n + 1 2 n is greater than or equal to 1 4 .
Solution Summary: The author analyzes whether the statement, "For each positive integer n, the sum of series is greater than or equal to 14" is valid.
A driver is traveling along a straight road when a buffalo runs into the street. This driver has a reaction time of 0.75 seconds. When the driver sees the buffalo he is traveling at 44 ft/s, his car can decelerate at 2 ft/s^2 when the brakes are applied. What is the stopping distance between when the driver first saw the buffalo, to when the car stops.
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
1. Given the vector field F(x, y, z) = -xi, verify the relation
1
V.F(0,0,0) = lim
0+ volume inside Se
ff F• Nds
SE
where SE is the surface enclosing a cube centred at the origin and having edges of length 2€. Then,
determine if the origin is sink or source.
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